Korteweg–De Vries equation

Summary

The Korteweg–De Vries equation, or KdV equation, is the nonlinear partial differential equation $u_t + u_{xxx} + 6 u u_x = 0$ for a real function $u(x,t)$. It models long waves of small amplitude in shallow water. It has solitary wave solutions that keep their shape, and it is the first equation that was solved by the inverse scattering transform. It has infinitely many conserved quantities and is a central example of a completely integrable system.